Bayesian model comparison based on expected posterior priors for discrete decomposable graphical models

The implementation of the Bayesian paradigm to model comparison can be problematic. In particular, prior distributions on the parameter space of each candidate model require special care. While it is well known that improper priors cannot be used routinely for Bayesian model comparison, we claim that in general the use of conventional priors (proper or improper) for model comparison should be regarded as suspicious, especially when comparing models having different dimensions. The basic idea is that priors should not be assigned separately under each model; rather they should be related across models, in order to acquire some degree of compatibility, and thus allow fairer and more robust comparisons. In this connection, the Expected Posterior Prior (EPP) methodology represents a useful tool. In this paper we develop a procedure based on EPP to perform Bayesian model comparison for discrete undirected decomposable graphical models, although our method could be adapted to deal also with Directed Acyclic Graph models. We present two possible approaches. One, based on imaginary data, requires to single-out a base-model, is conceptually appealing and is also attractive for the communication of results in terms of plausible ranges for posterior quantities of interest. The second approach makes use of training samples from the actual data for constructing the EPP. It is universally applicable, but has limited flexibility due to its inherent double-use of the data. The methodology is illustrated through the analysis of a 2 × 3 × 4 contingency table.